By extending the hydrodynamic chiral gravitational responses to non-zero torsion we computed the linear response Kubo formula with torsion and clarified the torsionful hydrodynamic variations, \(u_\mu , g_{\mu \nu }\)  vs. \(e^a_{\mu }, \omega ^a_{\mu b}\) . We identify a simple reason why the CVE contributes at linear order with the gravitational \(T^2\)  term: it is directly related to the mixed torsional graviational anomaly with coefficient \(t_{\chi } = d_{\chi }\) . This complements the earlier arguments with thermodynamic variations and background sources formulated in terms of \(T, \mu , u_i, g_{\mu \nu }\)  utilizing the mixed gravitational \(R^2\)  anomaly but without torsion. In particular, recent work on anomalous (non-)relativistic transport with vierbeins [1]}, [2]} did not take into account the presence of the Nieh-Yan anomaly terms, see however [3]}. The results we find with the relativistic Kubo formulas is in exact agreement with previous results, including the simple but physically compelling Landau level approach [4]}, [5]}.
